\section*{Problem 3}
Answer the following questions. Here, for any complex value $z , \bar { z }$ is the complex conjugate of $z$, arg $z$ is the argument of $z , | z |$ is the absolute value of $z$, and $i$ is the imaginary unit.

I. Sketch the region of $z$ on the complex plane that satisfies the following:

$$z \bar { z } + \sqrt { 2 } ( z + \bar { z } ) + 3 i ( z - \bar { z } ) + 2 \leq 0$$

II. Answer the following questions on the complex valued function $f ( z )$ below.

$$f ( z ) = \frac { z ^ { 2 } - 2 } { \left( z ^ { 2 } + 2 i \right) z ^ { 2 } }$$

\begin{enumerate}
  \item Find all the poles of $f ( z )$ as well as the orders and residues at the poles.
  \item By applying the residue theorem, find the value of the following integral $I _ { 1 }$. Here, the integration path $C$ is the circle on the complex plane in the counterclockwise direction which satisfies $| z + 1 | = 2$.
$$I _ { 1 } = \oint _ { C } f ( z ) \mathrm { d } z$$
\end{enumerate}

III. Answer the following questions.

\begin{enumerate}
  \item Let $g ( z )$ be a complex valued function, which satisfies
$$\lim _ { | z | \rightarrow \infty } g ( z ) = 0$$
for $0 \leq \arg z \leq \pi$.\\
Let $C _ { R }$ be the semicircle, with radius $R$, in the upper half of the complex plane with the center at the origin.\\
Show
$$\lim _ { R \rightarrow \infty } \int _ { C _ { R } } e ^ { i a z } g ( z ) \mathrm { d } z = 0$$
where $a$ is a positive real number.
  \item Find the value of the following integral, $I _ { 2 }$ :
$$I _ { 2 } = \int _ { 0 } ^ { \infty } \frac { \sin x } { x } \mathrm {~d} x$$
\end{enumerate}